Proposition 4

Let $A$ be a ring, $R$ an $A$-algebra (not necessarily commutative) and $(x_i)_{1 \leq i \leq n}$ a finite family of elements of $R$ which are pairwise permutable. If, for all $i$, $x_i$ is integral over $A[x_1, \ldots, x_{i-1}]$ (and in particular if all the $x_i$ are integral over $A$), then the subalgebra $A[x_1, \ldots, x_n]$ of $R$ is a finitely generated $A$-module.

We argue by induction on $n$, the proposition being just (E_{II}) for $n = 1$. The induction hypothesis implies that $B = A[x_1, \ldots, x_{n-1}]$ is a finitely generated A-module; as $x_n$ is integral over $B$, $B[x_n] = A[x_1, \ldots, x_n]$ is a finitely generated B-module and therefore also a finitely generated A-module (Algebra, Chapter II, § 1, no. 13, Proposition 25).